a1 Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 E-mail: email@example.com
a2 Durham Business School Durham University Durham, DH1 3LB, UK E-mail: firstname.lastname@example.org
a3 Department of Mathematics and Statistics, Lancaster University Management School, Bailrigg, Lancaster LA1 4YX, UK E-mail: email@example.com
a4 Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 E-mail: firstname.lastname@example.org
We consider a network of parallel service stations each modeled as a single-server queue. Each station serves its own dedicated customers as well as generic customers who are routed from a central controller. We suppose that the cost incurred by a customer is an increasing function of her time spent in the system. In a significant advance on most previous work, we do not require waiting costs to be convex, still less linear. With the objective of minimizing the long-run average waiting cost, we develop two heuristic routing policies, one of which is based on dynamic programming policy improvement and the other on Lagrangian relaxation. In developing the latter policy, we show that each station is “indexable” under mild conditions for customers’ waiting costs and also prove some structural results on the admission control problem that naturally arises as a result of the Lagrangian relaxation. We then test the performance of our heuristics in an extensive numerical study and show that the Lagrangian heuristic demonstrates a strong level of performance in a range of traffic conditions. In particular, it clearly outperforms both a greedy heuristic, which is a standard proposal in complex routing problems, and a recent proposal from the heavy traffic literature.